DIY impossible triangle. The paradoxical world of impossible objects

Penrose triangle- one of the main impossible figures, also known as impossible triangle And tribar.

Penrose triangle (in color)

Story

This figure became widely known after the publication of an article on impossible figures in the British Journal of Psychology by the English mathematician Roger Penrose in 1958. Also in this article, the impossible triangle was depicted in the most general form- V the form of three beams connected to each other at right angles. Influenced by this article in Dutch artist Maurits Escher created one of his famous lithographs "Waterfall".

3D print of a Penrose triangle

Sculptures

A 13-meter sculpture of an impossible triangle made of aluminum was erected in 1999 in Perth (Australia)

The same sculpture when changing the viewpoint

Other figures

Although it is quite possible to construct analogues of the Penrose triangle based on regular polygons, the visual effect from them is not so impressive. As the number of sides increases, the object simply appears bent or twisted.

see also

• Three rabbits (English) Three hares)

Illusionism - in a broad sense, is the name for a philosophical position regarding certain phenomena; for the way of considering such phenomena; in a narrow sense - this is the name for several specific philosophical theories.

The Cafe Wall Illusion is an optical illusion created by synergy. different levels neural mechanisms: retinal neurons and visual cortex neurons.

An impossible figure is one of the types of optical illusions, a figure that at first glance seems to be a projection of an ordinary three-dimensional object, upon careful examination of which contradictory connections of the elements of the figure become visible. An illusion is created of the impossibility of the existence of such a figure in three-dimensional space.

The Impossible Cube is an impossible figure invented by Escher for his lithograph Belvedere. This is a two-dimensional figure that superficially resembles the perspective of a three-dimensional cube, which is incompatible with a real cube. In the Belvedere lithograph, a boy sitting at the base of the building holds an impossible cube. A drawing of a similar Necker cube lies at his feet, while the building itself contains the same properties of an impossible cube.

The impossible cube borrows the ambiguity of the Necker cube, in which the edges are drawn as line segments, and which can be interpreted in one of two different three-dimensional orientations.

The impossible cube is usually drawn as a Necker cube, in which the edges (segments) are replaced by seemingly solid bars.

In the Escher lithograph, the top four joints of the bars and the top intersection of the bars correspond to one of two interpretations of the Necker cube, while the bottom four connections and the bottom intersection correspond to the other interpretation. Other variations of the impossible cube combine these properties in other ways. For example, one of the cubes in the figure contains all eight connections according to one interpretation of the Necker cube, and both intersections correspond to another interpretation.

The apparent solidity of the bars gives the impossible cube more visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. The illusion plays on the human eye's interpretation of a two-dimensional drawing as a three-dimensional object. Three-dimensional objects may seem impossible when viewed from a certain angle and either by cutting the object in the right place or by using altered perspective, but human experience with rectangular objects does impossible perception more likely than illusions in reality.

Other artists, including Jos De Mey, also painted works with the impossible cube.

A fabricated photograph of the supposedly impossible cube was published in the June 1966 issue of Scientific American, where it was called the "Frimish Cage." The impossible cube was featured on an Austrian postage stamp.

Blivet, also known as poyut or devil's pitchfork, is an inexplicable figure, optical illusion and an impossible figure. It seems that three cylindrical rods turn into two bars.

Oscar Rutersvärd (usual spelling of the surname in Russian-language literature; more correctly Reutersvärd), Swede. Oscar Reutersvärd (November 29, 1915, Stockholm, Sweden - February 2, 2002, Lund) - "father of the impossible figure", a Swedish artist who specialized in the image impossible figures, that is, those that can be depicted (given the inevitable violations of perspective when representing 3-dimensional space on paper), but cannot be created. One of his figures received further development as the "Penrose triangle" (1934). The work of Ruthersvard can be compared with the work of Escher, however, if the latter used impossible figures as “skeletons” for the image fantasy worlds, then Rutersvärd was only interested in the figures as such. During his life, Ruthersvard painted about 2,500 figures in isometric projection. Ruthersvard's books have been published in many languages, including Russian.

Maurits Cornelis Escher (Dutch: Maurits Cornelis Escher [ˈmʌu̯rɪts kɔrˈneːlɪs ˈɛʃər̥]; June 17, 1898, Leeuwarden, the Netherlands - March 27, 1972, Hilversum, the Netherlands) - a Dutch graphic artist. Known primarily for his conceptual lithographs, wood and metal engravings, in which he masterfully explored the plastic aspects of the concepts of infinity and symmetry, as well as the peculiarities of the psychological perception of complex three-dimensional objects, he is the most prominent representative of imp art.

Several impossible figures have been invented - a ladder, a triangle and an x-prong. These figures are actually quite real in a three-dimensional image. But when an artist projects volume onto paper, the objects seem impossible. The triangle, which is also called the “tribar,” has become a wonderful example of how the impossible becomes possible when you put in the effort.

All these figures are beautiful illusions. The achievements of human genius are used by artists who paint in the imp art style.

Nothing is impossible. This can be said about the Penrose triangle. This is a geometrically impossible figure, the elements of which cannot be connected. After all, the impossible triangle became possible. Swedish painter Oscar Reutersvärd introduced the world to the impossible triangle made of cubes in 1934. O. Reutersvard is considered the pioneer of this visual illusion. In honor of this event, this drawing was later printed on a Swedish postage stamp.

And in 1958, mathematician Roger Penrose published a publication in an English magazine about impossible figures. It was he who created the scientific model of illusion. Roger Penrose was an incredible scientist. He conducted research in the theory of relativity, as well as the fascinating quantum theory. He was awarded the Wolf Prize together with S. Hawking.

It is known that the artist Maurits Escher, under the impression of this article, painted his amazing job— lithograph “Waterfall”. But is it possible to make a Penrose triangle? How to do it, if possible?

Tribar and reality

Although the figure is considered impossible, making a Penrose triangle with your own hands is as easy as shelling pears. It can be made from paper. Origami lovers simply could not ignore the tribar and nevertheless found a way to create and hold in their hands a thing that previously seemed beyond the imagination of a scientist.

However, we are deceived by our own eyes when we look at the projection of a three-dimensional object from three perpendicular lines. The observer thinks he sees a triangle, although in fact he does not.

Geometry crafts

The tribar triangle, as stated, is not actually a triangle. The Penrose triangle is an illusion. Only at a certain angle does an object look like an equilateral triangle. However, the object in its natural form is 3 faces of a cube. In such an isometric projection, 2 angles coincide on the plane: the one closest to the viewer and the farthest.

The optical illusion, of course, quickly reveals itself as soon as you pick up this object. The shadow also reveals the illusion, since the shadow of the tribar clearly shows that the angles do not coincide in reality.

Tribar made of paper. Scheme

How to make a Penrose triangle with your own hands from paper? Are there any schematics for this model? Today, 2 layouts have been invented in order to fold such an impossible triangle. Basic geometry tells you exactly how to fold an object.

To fold a Penrose triangle with your own hands, you will need to allocate only 10-20 minutes. You need to prepare glue, scissors for several cuts and paper on which the diagram is printed.

From such a blank the most popular impossible triangle is obtained. The origami craft is not too difficult to make. Therefore, it will definitely work out the first time, even for a schoolchild who has just started studying geometry.

As you can see, it turns out to be a very nice craft. The second piece looks different and folds differently, but the Penrose triangle itself ends up looking the same.

Steps to create a Penrose triangle from paper.

Choose one of 2 blanks convenient for you, copy the file and print. Here we give an example of the second layout model, which is a little simpler.

The “Tribar” origami blank itself already contains all the necessary tips. In fact, instructions for the circuit are not required. It is enough just to download it onto a thick paper medium, otherwise it will be inconvenient to work and the figure will not work out. If you cannot immediately print on cardboard, then you need to attach the sketch to the new material and cut out the drawing along the contour. For convenience, you can fasten with paper clips.

What to do next? How to fold a Penrose triangle with your own hands step by step? You need to follow this action plan:

1. Let's direct reverse side scissors those lines where you need to bend, according to the instructions. Bend all the lines
2. We make cuts where necessary.
3. Using PVA, we glue together those scraps that are intended to hold the part together into a single whole.

The finished model can be repainted in any color, or you can take colored cardboard for work in advance. But even if the object is made of white paper, all the same, everyone who enters your living room for the first time will certainly be discouraged by such a craft.

Triangle drawing

How to draw a Penrose triangle? Not everyone likes to do origami, but many people love to draw.

To begin with, draw a regular square of any size. Then a triangle is drawn inside, the base of which is the bottom side of the square. A small rectangle is placed in each corner, all sides of which are erased; Only those sides that are adjacent to the triangle remain. This is necessary to ensure that the lines are straight. The result is a triangle with truncated corners.

The next stage is the image of the second dimension. A strictly straight line is drawn from the left side of the upper lower corner. The same line is drawn starting from the lower left corner, and is slightly not brought to the first line of the 2nd dimension. Another line is drawn from the right corner parallel to the bottom side of the main figure.

The final stage is to draw the third one inside the second dimension using three more small lines. Small lines start from the lines of the second dimension and complete the image of a three-dimensional volume.

Other Penrose figures

Using the same analogy, you can draw other shapes - a square or a hexagon. The illusion will be maintained. But still, these figures are no longer so amazing. Such polygons simply appear to be very twisted. Modern graphics allows you to make more interesting versions of the famous triangle.

In addition to the triangle, the Penrose Staircase is also world famous. The idea is to trick the eye, making it appear that a person is continuously rising upward when moving clockwise, and downwards when moving counterclockwise.

The continuous staircase is best known for its association with M. Escher’s painting “Ascent and Descend”. It is interesting that when a person walks all 4 flights of this illusory staircase, he invariably ends up back where he started.

There are also other objects known that mislead the human mind, such as the impossible block. Or a box made according to the same laws of illusion with intersecting edges. But all these objects have already been invented based on an article by a remarkable scientist - Roger Penrose.

Impossible triangle in Perth

The figure named after the mathematician is honored. A monument was erected to her. In 1999, in one of the cities of Australia (Perth), a large Penrose triangle made of aluminum was installed, which is 13 meters in height. Tourists enjoy taking pictures next to the aluminum giant. But if you choose a different angle for photography, the deception becomes obvious.

Dmitry Rakov

Our eyes cannot know
the nature of objects.
So don’t force it on them
delusions of reason.

Titus Lucretius Carus

The common expression “optical illusion” is inherently incorrect. The eyes cannot deceive us, since they are only an intermediate link between the object and the human brain. Optical illusion usually occurs not because of what we see, but because we unconsciously reason and involuntarily get mistaken: “the mind can look at the world through the eye, and not with the eye.”

One of the most spectacular areas of the artistic movement of optical art (op-art) is imp-art (impossible art), based on the depiction of impossible figures. Impossible objects are drawings on a plane (any plane is two-dimensional) depicting three-dimensional structures that are impossible to exist in the real three-dimensional world. The classic and one of the simplest figures is the impossible triangle.

In an impossible triangle, each angle is itself possible, but a paradox arises when we consider it as a whole. The sides of the triangle are directed both towards and away from the viewer, so its individual parts cannot form a real three-dimensional object.

Strictly speaking, our brain interprets a drawing on a plane as a three-dimensional model. Consciousness sets the “depth” at which each point of the image is located. Our ideas about the real world face a contradiction, some inconsistency, and we have to make some assumptions:

• straight 2D lines are interpreted as straight 3D lines;
• 2D parallel lines are interpreted as 3D parallel lines;
• acute and obtuse angles are interpreted as right angles in perspective;
• the outer lines are considered as the boundary of the form. This outer boundary is extremely important for constructing a complete image.

Human consciousness first creates a general image of an object, and then examines individual parts. Each angle is compatible with spatial perspective, but when reunited they form a spatial paradox. If you close any of the corners of the triangle, then the impossibility disappears.

History of impossible figures

Errors in spatial construction were encountered by artists even a thousand years ago. But the first to construct and analyze impossible objects is considered to be the Swedish artist Oscar Reutersvärd, who in 1934 drew the first impossible triangle, consisting of nine cubes.

		"Moscow", graphics (mascara, pencil), 50x70 cm, 2003

Independent of Reuters, English mathematician and physicist Roger Penrose rediscovers the impossible triangle and publishes an image of it in a British psychology journal in 1958. The illusion uses “false perspective.” Sometimes this perspective is called Chinese, since a similar method of drawing, when the depth of the drawing is “ambiguous,” was often found in the works of Chinese artists.

In the "Three Snails" drawing, the small and large cubes are not oriented in a normal isometric projection. The smaller cube is adjacent to the larger one on the front and back sides, which means, following three-dimensional logic, it has the same dimensions of some sides as the larger one. At first, the drawing seems to be a real representation of a solid body, but as analysis proceeds, the logical contradictions of this object are revealed.

The "Three Snails" drawing continues the tradition of the second famous impossible figure - the impossible cube (box).

"IQ", graphics (mascara, pencil), 50x70 cm, 2001		"Up and down", M. Escher

A combination of various objects can also be found in the not entirely serious drawing “IQ” (intelligence quotient). Interestingly, some people do not perceive impossible objects because their minds are unable to identify flat pictures with three-dimensional objects.

Donald E. Simanek has suggested that understanding visual paradoxes is one of the hallmarks of the kind of creativity that the best mathematicians, scientists and artists possess. Many works with paradoxical objects can be classified as “intellectual mathematical games”. Modern science speaks of a 7-dimensional or 26-dimensional model of the world. Such a world can only be modeled using mathematical formulas; humans simply cannot imagine it. This is where impossible figures come in handy. From a philosophical point of view, they serve as a reminder that any phenomena (in system analysis, science, politics, economics, etc.) should be considered in all complex and non-obvious relationships.

A variety of impossible (and possible) objects are presented in the painting "Impossible Alphabet".

A third popular impossible figure is the incredible staircase created by Penrose. You will continuously either ascend (counterclockwise) or descend (clockwise) along it. Penrose's model formed the basis of the famous painting by M. Escher "Up and Down" ("Ascending and Descending").

There is another group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork".

If you carefully study the picture, you will notice that three teeth gradually turn into two on a single base, which leads to a conflict. We compare the number of teeth above and below and come to the conclusion that the object is impossible.

Is there any greater benefit from impossible drawings than mind games? Some hospitals deliberately hang up pictures of impossible objects, since looking at them can keep patients busy for a long time. It would be logical to hang such drawings at ticket offices, police stations and other places where waiting in line sometimes lasts an eternity. The drawings could act as sort of “chronophages”, i.e. time wasters.

Also known as impossible triangle And tribar.

Story

This figure became widely known after the publication of an article on impossible figures in the British Journal of Psychology by the English mathematician Roger Penrose in 1958. In this article, the impossible triangle was depicted in its most general form - in the form of three beams connected to each other at right angles. Influenced by this article, the Dutch artist Maurits Escher created one of his famous lithographs, “Waterfall”.

Sculptures

A 13-meter sculpture of an impossible triangle made of aluminum was erected in 1999 in Perth (Australia)

Deutsches Technikmuseum Berlin February 2008 0004.JPG

The same sculpture when changing the viewpoint

Other figures

Although it is quite possible to construct analogues of the Penrose triangle based on regular polygons, the visual effect from them is not so impressive. As the number of sides increases, the object simply appears bent or twisted.

see also

• Three rabbits (English) Three hares )

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An excerpt characterizing the Penrose Triangle